Solve 2x/x^2+x-6-2/x+3+5/x+9

$$\frac { 2 x } { x ^ { 2 } + x - 6 } - \frac { 2 } { x + 3 } + \frac { 5 } { x + 9 }$$

Evaluate

$$\frac{5x^{2}+9x+6}{\left(x-2\right)\left(x+3\right)\left(x+9\right)}$$

Short Solution Steps

Factor x^{2}+x-6.

$$\frac{2x}{\left(x-2\right)\left(x+3\right)}-\frac{2}{x+3}+\frac{5}{x+9}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x+3\right) and x+3 is \left(x-2\right)\left(x+3\right). Multiply \frac{2}{x+3} times \frac{x-2}{x-2}.

$$\frac{2x}{\left(x-2\right)\left(x+3\right)}-\frac{2\left(x-2\right)}{\left(x-2\right)\left(x+3\right)}+\frac{5}{x+9}$$

Since \frac{2x}{\left(x-2\right)\left(x+3\right)} and \frac{2\left(x-2\right)}{\left(x-2\right)\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{2x-2\left(x-2\right)}{\left(x-2\right)\left(x+3\right)}+\frac{5}{x+9}$$

Do the multiplications in 2x-2\left(x-2\right).

$$\frac{2x-2x+4}{\left(x-2\right)\left(x+3\right)}+\frac{5}{x+9}$$

Combine like terms in 2x-2x+4.

$$\frac{4}{\left(x-2\right)\left(x+3\right)}+\frac{5}{x+9}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-2\right)\left(x+3\right) and x+9 is \left(x-2\right)\left(x+3\right)\left(x+9\right). Multiply \frac{4}{\left(x-2\right)\left(x+3\right)} times \frac{x+9}{x+9}. Multiply \frac{5}{x+9} times \frac{\left(x-2\right)\left(x+3\right)}{\left(x-2\right)\left(x+3\right)}.

$$\frac{4\left(x+9\right)}{\left(x-2\right)\left(x+3\right)\left(x+9\right)}+\frac{5\left(x-2\right)\left(x+3\right)}{\left(x-2\right)\left(x+3\right)\left(x+9\right)}$$

Since \frac{4\left(x+9\right)}{\left(x-2\right)\left(x+3\right)\left(x+9\right)} and \frac{5\left(x-2\right)\left(x+3\right)}{\left(x-2\right)\left(x+3\right)\left(x+9\right)} have the same denominator, add them by adding their numerators.

$$\frac{4\left(x+9\right)+5\left(x-2\right)\left(x+3\right)}{\left(x-2\right)\left(x+3\right)\left(x+9\right)}$$

Do the multiplications in 4\left(x+9\right)+5\left(x-2\right)\left(x+3\right).

$$\frac{4x+36+5x^{2}+15x-10x-30}{\left(x-2\right)\left(x+3\right)\left(x+9\right)}$$

Combine like terms in 4x+36+5x^{2}+15x-10x-30.

$$\frac{9x+6+5x^{2}}{\left(x-2\right)\left(x+3\right)\left(x+9\right)}$$

Expand \left(x-2\right)\left(x+3\right)\left(x+9\right).

$$\frac{9x+6+5x^{2}}{x^{3}+10x^{2}+3x-54}$$

Differentiate w.r.t. x

$$-\frac{5x^{4}+18x^{3}+93x^{2}+660x+504}{\left(\left(x-2\right)\left(x+3\right)\left(x+9\right)\right)^{2}}$$